Theorem exmidontri2or 7241

Description: Ordinal trichotomy is equivalent to excluded middle. (Contributed by Jim Kingdon, 26-Aug-2024.)

Assertion

Ref	Expression
exmidontri2or	|- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Distinct variable group:   x, y

Proof of Theorem exmidontri2or

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidontriim 7223	. . 3 |- (EXMID -> A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) )
2		onelss 4387	. . . . . . . 8 |- ( y e. On -> ( x e. y -> x C_ y ) )
3	2	adantl 277	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> x C_ y ) )
4		orc 712	. . . . . . 7 |- ( x C_ y -> ( x C_ y \/ y C_ x ) )
5	3, 4	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x e. y -> ( x C_ y \/ y C_ x ) ) )
6		eqimss 3209	. . . . . . . 8 |- ( x = y -> x C_ y )
7	6, 4	syl 14	. . . . . . 7 |- ( x = y -> ( x C_ y \/ y C_ x ) )
8	7	a1i 9	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( x = y -> ( x C_ y \/ y C_ x ) ) )
9		onelss 4387	. . . . . . . 8 |- ( x e. On -> ( y e. x -> y C_ x ) )
10	9	adantr 276	. . . . . . 7 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> y C_ x ) )
11		olc 711	. . . . . . 7 |- ( y C_ x -> ( x C_ y \/ y C_ x ) )
12	10, 11	syl6 33	. . . . . 6 |- ( ( x e. On /\ y e. On ) -> ( y e. x -> ( x C_ y \/ y C_ x ) ) )
13	5, 8, 12	3jaod 1304	. . . . 5 |- ( ( x e. On /\ y e. On ) -> ( ( x e. y \/ x = y \/ y e. x ) -> ( x C_ y \/ y C_ x ) ) )
14	13	ralimdva 2544	. . . 4 |- ( x e. On -> ( A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. y e. On ( x C_ y \/ y C_ x ) ) )
15	14	ralimia 2538	. . 3 |- ( A. x e. On A. y e. On ( x e. y \/ x = y \/ y e. x ) -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
16	1, 15	syl 14	. 2 |- (EXMID -> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )
17		ontri2orexmidim 4571	. . . 4 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> DECID z = { (/) } )
18	17	adantr 276	. . 3 |- ( ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) /\ z C_ { (/) } ) -> DECID z = { (/) } )
19	18	exmid1dc 4200	. 2 |- ( A. x e. On A. y e. On ( x C_ y \/ y C_ x ) -> EXMID )
20	16, 19	impbii 126	1 |- (EXMID <-> A. x e. On A. y e. On ( x C_ y \/ y C_ x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   \/ w3o 977   = wceq 1353   e. wcel 2148   A.wral 2455   C_ wss 3129   (/)c0 3422   {csn 3592  EXMIDwem 4194   Oncon0 4363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-tr 4102  df-exmid 4195  df-iord 4366  df-on 4368  df-suc 4371

This theorem is referenced by:  onntri52  7242